The roots are found with the following two statements. We have denoted the polynomial as p1, and the roots as roots_ p1.

Transcription

1 Part II Lesson 10 Numerical Analysis Finding roots of a polynomial In MATLAB, a polynomial is expressed as a row vector of the form [an an 1 a2 a1 a0]. The elements ai of this vector are the coefficients of the polynomial in descending order. We must include terms whose coefficients are zero. We can find the roots of any polynomial with the roots (p) function where p is a row vector containing the polynomial coefficients in descending order. Exercise 1.!! Find the roots of the polynomial There is no cube term; therefore, we must enter zero as its coefficient. The roots are found with the statements below where we have defined the polynomial as p2, and the roots of this polynomial as roots_ p2. The result indicates that this polynomial has three real roots, and two complex roots. p2=[ ] p2 =! 1! -7! 0! 16! 25! 52 roots_ p2=roots(p2) roots_p2 =!! !! ! ! i! i Exercise 2.!! Find the roots of the polynomial p1(x) = x 4 10 x x 2 50x +24 The roots are found with the following two statements. We have denoted the polynomial as p1, and the roots as roots_ p1. p1=[ ] % Specify the coefficients of p1(x) p1 =! roots_ p1=roots(p1) % Find the roots of p1(x) roots_p1 =! ! ! ! From the above results we observe that MATLAB displays the polynomial coefficients as a row vector, and the roots as a column vector.

2 Exercise 3. Solving Quadratic Equations You can solve equations involving variables with solve or fzero. To find the solutions of the quadratic equation, type the following in command prompt >> solve( xˆ2-2*x - 4 = 0 ) ans =! [ 5^(1/2)+1]! [ 1-5^(1/2)] Note that the equation to be solved is specified as a string; that is, it is surrounded by single quotes. The answer consists of the exact (symbolic) solutions

3 LESSON 11 Matrices Exercise 1. Compute A+B and A B given that Solution: and Check with MATLAB in command prompt: Exercise 2. Transpose of a matrix The transpose of a matrix A, denoted as A T, is the matrix that is obtained when the rows and columns of matrix A are interchanged. For example, if In MATLAB we use the apostrophe ( ) symbol to denote and obtain the transpose of a matrix. Thus, for the above example,

4 A' % Display the transpose of A Exercise 3. (a) Determinants

5 Exercise 3. (b) Compute the determinant of A using the elements of the first row. Exercise 4. Cramer s Rule

6 We will verify with MATLAB as follows

7 Exercise 5. Solution of Simultaneous Equations with Matrices Consider the relation, AX = B where A and B are matrices whose elements are known, and X is a matrix (a column vector) AX = B whose elements are the unknowns. We assume that A and X are conformable for multiplication. Multiplication of both sides of AX = B by A 1 yields: Therefore, we can use the above equation to solve any set of simultaneous equations that have solutions. We will refer to this method as the inverse matrix method of solution of simultaneous equations. Given the system of equations Compute the unknown s x1, x2, and x3 using the inverse matrix method. Solution: In matrix form, the given set of equations is AX = B where

8 and by X = A -1 B we obtain

9 Exercise 6. Finding Eigen values and Eigen vectors 1. Find eigen vales and eigen vectors of the following 3 x 3 matrix using command window >> A=[5-3 2; ; 4 2-9]; >> eig(a) ans = >> [eigvec,eigval]=eig(a) eigvec = eigval = Find the eigen value and eigen vector for the following 2 X 2 matrix using command window >> A=[ ; ]; >> [eigvec,eigval]=eig(a) eigvec = eigval =

10 Lesson 12 Ordinary Differential Equations Exercise 1. Solving a first order ordinary differential equation Solve the first order linear differential equation with the initial conditions x(0) = 0. Step 1: Write the equation(s) as a system of first-order equations: Step 2: Write a function to compute the new derivatives and save it as an M-File named simpode.m function xdot = simpode(t,x); %SIMPODE: Computes xdot = x + t %Call syntax: xdot = simpode(t,x); xdot = x + t;! Step 3: Use ode23 to compute solution in the command window >>tspan = [0 2];!! %Specify time span! >>x0 = 0;!!! %Specify x0! >>[t,x] = ode23( simpode,tspan,x0); % Now executes ode23! >>plot(t,x)!! %plot t vs. x! >>xlabel( t )!! %label x-axis! >>ylabel ( x )!! %label y-axis the plot generated by the above commands are shown below

11 Excercise 2. Solve the simple ODE with the initial condition x(0)=1 Step 1: Write the equation(s) as a system of first-order equations: Step 2: Write a function to compute the new derivatives and save it as an M-File named simpode1.m function xdot = simpode(t,x); %SIMPODE: Computes xdot = x + t %Call syntax: xdot = simpode(t,x); xdot = -10*x;! Step 3: Use ode23 to compute solution in the command window >>tspan = [0 1];! %Specify time span! >>x0 = 1;!!! %Specify x0! >>[t,x] = ode23( simpode1,tspan,x0); % Now executes ode23! >>plot(t,x)!! %plot t vs. x! >>xlabel( t )!! %label x-axis! >>ylabel ( x )!! %label y-axis the plot generated by the above commands are shown below